Question: $f(x)=\dfrac{6}{1+x^3}$ Find a power series for $f$. Choose 1 answer: Choose 1 answer: (Choice A) A $6-6{{x}^{3}}+6{{x}^{6}}+\ldots +6{{\left(- {{x}^{3}} \right)}^{n}}+\ldots$ (Choice B) B $6+6{{x}^{3}}+6{{x}^{6}}+\ldots +6{{\left( {{x}^{3}} \right)}^{n}}+\ldots$ (Choice C) C $6-216{{x}^{3}}+1296{{x}^{6}}+\ldots +{{\left(-6 {{x}^{3}} \right)}^{n}}+\ldots$ (Choice D) D $6-6{{x}^{3}}-6{{x}^{6}}+\ldots -6{{\left( {{x}^{3}} \right)}^{n}}+\ldots$
Solution: This is a geometric series with first term $a\text{ }=\text{ }6$ and common ratio $r=-{{x}^{3}}$. Therefore, the series is as follows. $6\text{ }-\text{ }6{{x}^{3}}+\text{ }6{{x}^{6}}-\text{ }6{{x}^{9}}+\ldots +\text{ }6{{\left(- {{x}^{3}} \right)}^{n}}+\ldots $